Question $\begin{array}{l} y<-4 x+6 \\ y>\frac{1}{4} x+4 \end{array}$
Answer $(3,9)$ $(-2,-4)$ (9.5) $(-7.9)$ DeltaMath - Googl...

Studdy Solution

STEP 1

1. We are given two inequalities that define a region in the coordinate plane.
2. We need to determine if the points $(3,9)$ , $(-2,-4)$ , $(9,5)$ , and $(-7,9)$ satisfy both inequalities.

STEP 2

1. Analyze the inequality $y < -4x + 6$ .
2. Analyze the inequality $y > \frac{1}{4}x + 4$ .
3. Test each point against both inequalities.
4. Determine which points satisfy both inequalities.

STEP 3

Analyze the inequality $y < -4x + 6$ :
This inequality represents the region below the line $y = -4x + 6$ . For a point $(x, y)$ to satisfy this inequality, its $y$ -coordinate must be less than the value given by substituting its $x$ -coordinate into the equation of the line.

STEP 4

Analyze the inequality $y > \frac{1}{4}x + 4$ :
This inequality represents the region above the line $y = \frac{1}{4}x + 4$ . For a point $(x, y)$ to satisfy this inequality, its $y$ -coordinate must be greater than the value given by substituting its $x$ -coordinate into the equation of the line.

STEP 5

Test each point against both inequalities:
- For $(3, 9)$ : - Check $y < -4x + 6$ : $9 < -4(3) + 6 = -12 + 6 = -6$ (False) - Check $y > \frac{1}{4}x + 4$ : $9 > \frac{1}{4}(3) + 4 = \frac{3}{4} + 4 = 4.75$ (True) - Result: Does not satisfy both.
- For $(-2, -4)$ : - Check $y < -4x + 6$ : $-4 < -4(-2) + 6 = 8 + 6 = 14$ (True) - Check $y > \frac{1}{4}x + 4$ : $-4 > \frac{1}{4}(-2) + 4 = -0.5 + 4 = 3.5$ (False) - Result: Does not satisfy both.
- For $(9, 5)$ : - Check $y < -4x + 6$ : $5 < -4(9) + 6 = -36 + 6 = -30$ (False) - Check $y > \frac{1}{4}x + 4$ : $5 > \frac{1}{4}(9) + 4 = 2.25 + 4 = 6.25$ (False) - Result: Does not satisfy both.
- For $(-7, 9)$ : - Check $y < -4x + 6$ : $9 < -4(-7) + 6 = 28 + 6 = 34$ (True) - Check $y > \frac{1}{4}x + 4$ : $9 > \frac{1}{4}(-7) + 4 = -1.75 + 4 = 2.25$ (True) - Result: Satisfies both.

STEP 6

Determine which points satisfy both inequalities:
The point $(-7, 9)$ satisfies both inequalities.
The point that satisfies both inequalities is $(-7, 9)$ .

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